Three Valued Logic
Another thing that has been bothering me greatly about Zeroth Order Logic is that it makes a closed world assumption: what isn't stated is assumed to be false.
There is much that we don't explicitly state (often for brevity, but sometimes because we genuinely do not know), so finding a way to reason over propositions using an open world assumption is needed: what isn't stated should be assumed as maybe.
There are many variations of three value logics, but all of them assume that propositions can take three values:
- true,
- false and
- maybe.
Connectives
Kleene's K3 and Priest's P3 logic have the same truth values for the unary connector:
| p | ~p |
|---|---|
| F | T |
| T | F |
| ? | ? |
And binary connectors:
| p | q | p or q | p and q | p => q | p <=> q | p xor q |
|---|---|---|---|---|---|---|
| F | F | F | F | T | T | F |
| F | T | T | F | T | F | T |
| F | ? | ? | F | T | ? | ? |
| T | F | T | F | F | F | T |
| T | T | T | T | T | T | F |
| T | ? | T | ? | ? | ? | ? |
| ? | F | ? | F | ? | ? | ? |
| ? | T | T | ? | T | ? | ? |
| ? | ? | ? | ? | ? | ? | ? |
Notably, only lines 1, 2, 4 and 5 are exactly the ones in zeroth order logic, since p and q don't assume values of maybe.
Tautologies
A logical formula is considered a tautology if it evaluates to a designated truth value in every possible interpretation.
K3 and P3, however, differ in how tautologies are defined. In K3, only true is a designed truth value, whereas in P3, both true and maybe are. So, K3 does not have any tautologies, while P3 has the same tautologies as classical two-valued logic.
For example, p or ~p is a tautology in P3 but it is not in K3.
| p | ~p | p or ~p |
|---|---|---|
| F | T | T |
| T | F | T |
| ? | ? | ? |
Because of that, P3 has the same tautologies as classical two-valued logic whereas K3 has none.
Equivalences
Two propositions p and q are logically equivalent when p <=> q is a tautology (i.e. true regardless of the values of p and q).
For example, take the double negation tautology: ~~p <=> p is true or maybe regardless of the value of p:
| p | ~p | ~~p | ~~p <=> p |
|---|---|---|---|
| F | T | F | T |
| T | F | T | T |
| ? | ? | ? | ? |
Do you think the other equivalences of two-valued logic are kept?
Implications
Implications are tautologies where you establish a sufficiency => but not a necessity <=.
For example, from the following premises:
pandp => q
Can we conclude q? That is, does (p and p => q) => q?
If modus ponens was a tautology in P3, you'd be able to enumerate all possible interpretation of p and q and verify that it is true or maybe for any permutation.
| p | q | p => q | (p and p => q) | (p and p => q) => q |
|---|---|---|---|---|
| F | F | T | F | T |
| F | T | T | F | T |
| F | ? | T | F | T |
| T | F | F | F | T |
| T | T | T | T | T |
| T | ? | ? | ? | ? |
| ? | F | ? | F | T |
| ? | T | T | ? | T |
| ? | ? | ? | ? | ? |
Rules of Deduction
Recall that in the two-valued propositional logic the implication could also be written as "not (p and not q)" and "(not a) or b":
| p | q | p => q | not (p and not q) | (not a) or b |
|---|---|---|---|---|
| F | F | T | T | T |
| F | T | T | T | T |
| T | F | F | F | F |
| T | T | T | T | T |
But if we take that to the three-valued logic, we get:
| p | q | p => q | not (p and not q) | (not p) or q |
|---|---|---|---|---|
| F | F | T | T | T |
| F | T | T | T | T |
| F | ? | T | T | T |
| T | F | F | F | F |
| T | T | T | T | T |
| T | ? | ? | ? | ? |
| ? | F | ? | ? | ? |
| ? | T | T | T | T |
| ? | ? | ? | ? | ? |
Which, by my calculation, seems to match up.
It is said, though, that modus ponens (and others) cannot be used as a rule of deduction in P3 (which seems embarrasingly limiting)*, which makes me wonder if we we defined implications imprecisely.
* same here
WDYT?
So, more to follow!